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Results tagged with quantum-groups
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user 703
Questions about algebraic structures known as quantum groups, and their categories of representations. Quasitriangular Hopf algebras and their Drinfel'd twists, triangular Hopf algebras, $C^\star$ quantum groups, h-adic quantum groups, various semisimplified categories at roots of unity which are called "quantum groups", bicrossproduct quantum groups, and quantum groups coming from braided tensor categories.
10
votes
Is the nc torus a quantum group?
The $C^*$-algebra versions are treated in this paper by Piotr Soltan:
http://arxiv.org/abs/0904.3019
The abstract reads: We prove that some well known compact quantum spaces like quantum tori and so …
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1
vote
Drinfeld's equivalence of quantized function algebras and quantized universal enveloping alg...
I don't have the text of Drinfeld's address near to hand, but the standard way I know to do this is to take the subalgebra of the (finite) dual generated by the matrix coefficients of the irreducible …
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1
vote
Explicit Coquasi-Triangular Quantised Coordinate Algebra of a Complex Semi-Simple Lie Group?
The coefficients of $R$ are essentially the coefficients of the braiding of the vector representation of $U_q(\mathfrak{g})$. So, more or less, you are asking for a general formula in terms of Cartan …
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2
votes
Generators of the Quantum Coordinate Algebras and Quantized Enveloping Algebra Representations
For your first question, the answer is yes, as Casteels pointed out in the comments. The reason is that, for $\mathfrak{sl}_N$, every finite-dimensional irreducible representation appears as a subrep …
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2
votes
Are the Drinfeld compact quantum groups simply connected ?
This is a negative answer to your question, or at least a partially negative one. I don't know if you've thought about things in this way, but there is a naive way to formulate the idea of the fundam …
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7
votes
Accepted
Is there a quantum Hermite reciprocity?
There is in fact a reasonable way to define quantum analogues of symmetric and exterior powers of a finite-dimensional representation of $U_q(\mathfrak{g})$. Let $V$ be such a representation, and let …
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7
votes
0
answers
209
views
Does the braid group act faithfully on the quantized enveloping algebra?
Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$, and let $U_q(\mathfrak{g})$ be some incarnation of the quantized universal enveloping algebra of $\mathfrak{g}$, where …
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7
votes
1
answer
287
views
Real forms of Drinfeld-Jimbo quantum groups
A real form of a Hopf algebra $H$ over $\mathbb{C}$ is defined to be a $\ast$-structure on $H$ which is compatible with the coproduct. Compatibility of the $\ast$-structure with the counit and antipo …
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2
votes
History of the Odd Dimensional Quantum Spheres
(1) I think Podles only introduced the quantum 2-spheres. His paper is linked from MathSciNet, so you should be able to get it if you have access to ams.org. I think the higher-dimensional spheres …
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5
votes
Accepted
Generators of the Odd Dimensional Quantum Spheres
This is shown in the book Quantum Groups and Their Representations, by Klimyk and Schmudgen. The result you ask for is Proposition 63 in Chapter 11. I'd expand more upon this but I have to give a ta …
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2
votes
Non-Drinfeld–Jimbo deformations and finite quantum groups
I do not know of a general method for quantizing the group algebra of a finite group. However, there is a way to do it for Coxeter groups (finite or not): the result is called an Iwahori-Hecke algebr …
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14
votes
Accepted
Relationship between "different" quantum deformations
There is certainly a way to quantize the algebra of functions on a Lie group in a way that is compatible with the $q$-deformation of the universal enveloping algebra of its Lie algebra. The standard …
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8
votes
Accepted
Reference for the Hecke relation for the universal R-matrix
For the Drinfeld-Jimbo quantum universal enveloping algebras, see Proposition 24 of Chapter 8 in the book Quantum Groups and Their Representations, by Klimyk and Schmudgen. This relation is just in t …
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4
votes
Accepted
R-matrices, crystal bases, and the limit as q -> 1
I never found a precise reference for the statement about the R-matrix, so I ended up writing it up myself. The precise statements and proofs can be found in $\S 4.1$ of my paper with Alex Chirvasitu …
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3
votes
An inner product that makes the R-matrix unitary
I'm pretty late to the party here, and Ben, it seems that you already have a satisfactory answer to your question, but I thought for the sake of completeness I would just post this in case anybody stu …
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11
votes
1
answer
761
views
R-matrices, crystal bases, and the limit as q -> 1
I am seeking references for precise statements and rigorous proofs of some facts about the actions of quantum root vectors and $R$-matrices on crystal bases for finite-dimensional representations of q …
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11
votes
1
answer
673
views
Unitary representations of Quantum Groups
Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra and let $U_q(\mathfrak{g})$ be some incarnation of the quantized universal enveloping algebra of $\mathfrak{g}$; here I am assumin …
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3
votes
Deformation quantization of a closed Riemann surface with genus >1
See the paper Quantization of Multiply Connected Manifolds, by Eli Hawkins. arXiv link.
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